17 problems

Let $G$ be a group, let $\Bbbk$ be a field of characteristic $l$, and let $(C,\mathcal{E})\in\mathscr{I}(G)$ be a $0$-cuspidal pair. A prime $l$ is rather good for $G$ if it is goo…

Let $G_0$ be any connected unipotent group over ${\mathbb F}_q$, let $G=G_0\otimes_{{\mathbb F}_q}{\mathbb F}$ with Frobenius $\operatorname{Fr}$, and let…

Let $H$ be an algebraic group over the ground field, and call it easy if every $h$ lies in the neutral connected component of its centralizer. The easy-group characterization conje…

Let $G$ be a connected unipotent algebraic group, let $e$ be a minimal idempotent of ${\mathscr D}_G(G)$, and let its functional dimension be $d_e=(\dim G-n_e)/2$. The integrality…

The modularity conjecture. The subcategory ${\mathcal M}_e$ is closed under convolution and hence is a monoidal category with unit $e$ if the preceding perversity conjecture holds;…

Let $G$ be a connected unipotent algebraic group, let $e$ be a minimal idempotent of ${\mathscr D}_G(G)$, let ${\mathcal M}_e^{perv}$ be the full subcategory of perverse objects in…

Let ${\mathcal C}_\zeta^{{\underline c}_f}$ be the monoidal category associated with a finite two-sided cell ${\underline c}_f\subset W_f(\zeta)$, and let…

Let $G$ be a reductive group, let $\xi=(M,\lambda,L,\mathcal{O},\mathcal{L})$ be a polar-cuspidal datum for $G$, and let $\mathbf{Q}\subset LM$ be a parahoric subgroup containing…

Let $G$ be a connected reductive group over the maximal unramified extension of a non-archimedean local field, let $G_r$ be the truncated parahoric group scheme associated to a poi…

Let $G$ be the connected reductive group with involution $\theta$, let $K=G^\theta$, and let $\mathfrak g_1$ be the degree-one component of the associated graded Lie algebra. For a…

Let ${\mathfrak{g}}_1$ be the degree-one component of one of the stably graded Lie algebras considered here, let $K$ act on ${\mathfrak{g}}_1$, and let ${\mathfrak{g}}_1^{rs}$ be i…

Let $\Sigma$ be a compact Riemann surface, let $G$ be a reductive group, and let $\check{G}$ be its Langlands dual group. Write $\operatorname{Bun}_G(\Sigma)$ for the moduli stack…

Let $G$ be a complex reductive group, let $E$ be an elliptic curve, and let $\mathcal H_G$ be the affine Hecke category. Let $\mathcal Shv_{\mathcal N}(Bun_G(E))$ denote the catego…

Let $G^1$ be the relevant component, let $\overline{G^1}^{ss}$ be its semistable locus, and let $S'$ be the simple perverse sheaf on $\overline{G^1}^{ss}$ obtained from the semisim…

Let $A'$ be a character sheaf on $G^1$, let $I$ be the set indexing the relevant standard Levi subgroups, and let $J\subset I$. Write $Z_{J,\delta;1}$ for the corresponding boundar…

Let $D$ be a vector space, and let $V$ and $W$ be vector spaces with $\dim(V)+\dim(W)=\dim(D)$. Let $\CG\in\calC(\operatorname{GL}(V))$ and $\CG'\in\calC(\operatorname{GL}(W))$, an…

Mirabolic character-sheaf formula conjecture.